Foundations of Vectors and the Cartesian Plane: Essential Pre-Calculus Concepts

1. The Set of Real Numbers

Definition and Classification of Numbers

The real numbers form the foundational set for calculus and vector analysis. Understanding their structure is essential for further study in mathematics and physics.

• Integers: The set of numbers including positive whole numbers (1, 2, 3, ...), negative whole numbers (−1, −2, −3, ...), and zero (0).

• Rational Numbers: Any number that can be expressed as a ratio of two integers, $x = \frac{m}{n}$, where $m, n$ are integers and $n \neq 0$.

• Irrational Numbers: Numbers that cannot be written as a ratio of two integers. Examples include $\sqrt{2}$, $\pi$, and $e$.

• Real Numbers: The union of rational and irrational numbers. Unless otherwise specified, 'number' refers to a real number.

Example: $\frac{1}{2}$, $-\frac{3}{7}$, and $\frac{9}{-8} = -\frac{9}{8}$ are rational; $\sqrt{2}$ and $\pi$ are irrational.

2. The Number Line

Constructing and Interpreting the Number Line

The number line provides a geometric representation of real numbers, associating each number with a unique point.

• Choose a unit of length and mark equally spaced points on a line.

• Assign the number 0 to a reference point O (the origin).

• For a point P, the distance $|OP|$ is always positive.

• If P is to the right of O, $x = |OP|$; if to the left, $x = -|OP|$.

• Given a number $x$, the corresponding point is $|x|$ units from O: right if $x > 0$, left if $x < 0$, and at O if $x = 0$.

• Absolute Value: $|x|$ denotes the distance from O to the point representing $x$.

Example: $x = 3$ is 3 units right of O; $x = -3$ is 3 units left of O.

3. The Cartesian Coordinate System

Ordered Pairs and the Plane

The Cartesian coordinate system extends the number line to two dimensions, allowing each point in the plane to be represented by an ordered pair $(x, y)$.

• Construct two perpendicular number lines (axes) intersecting at the origin O.

• The horizontal axis is the x-axis; the vertical is the y-axis.

• Each point P in the plane is associated with coordinates $(x, y)$, where $x$ is the horizontal and $y$ the vertical component.

• The order of coordinates matters: $(1, 2) \neq (2, 1)$.

• The axes divide the plane into four quadrants:

  • Quadrant I: $x > 0$, $y > 0$

  • Quadrant II: $x < 0$, $y > 0$

  • Quadrant III: $x < 0$, $y < 0$

  • Quadrant IV: $x > 0$, $y < 0$

• If $x = 0$, the point lies on the y-axis; if $y = 0$, on the x-axis.

Example: P($-4$, 2) is in Quadrant II; Q(2, $-3$) is in Quadrant IV.

4. Two-Dimensional Vectors

Definition and Representation

A two-dimensional vector is an ordered pair $\mathbf{a} = \langle a_1, a_2 \rangle$, representing both magnitude and direction in the plane.

• Components: $a_1$ (horizontal), $a_2$ (vertical).

• Graphically, a vector is a directed line segment (arrow) from point P($x$, $y$) to Q($x + a_1$, $y + a_2$).

4.1 The Magnitude of a Vector

• The magnitude (length) of $\mathbf{a} = \langle a_1, a_2 \rangle$ is given by:

$\|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2}$

• Derived from the Pythagorean theorem.

4.2 The Direction of a Vector

• The direction is determined by the signs and values of $a_1$ and $a_2$.

• The head of the vector is $|a_1|$ units right/left and $|a_2|$ units up/down from the tail, depending on the sign.

• Vectors with the same magnitude and direction are equivalent, regardless of their position in the plane.

Example: $\mathbf{a} = \langle 2, 1 \rangle$ can be represented by arrows from any point P($x$, $y$) to Q($x+2$, $y+1$).

5. Vector Algebra

5.1 The Zero Vector

• The zero vector $\mathbf{0} = \langle 0, 0 \rangle$ has zero magnitude and no direction.

5.2 Vector Equality

• Vectors $\mathbf{a} = \langle a_1, a_2 \rangle$ and $\mathbf{b} = \langle b_1, b_2 \rangle$ are equal if and only if $a_1 = b_1$ and $a_2 = b_2$.

5.3 Vector Addition

• The sum of two vectors is:

$\mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle$

• Commutative: $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$

• Associative: $(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c})$

• Identity: $\mathbf{a} + \mathbf{0} = \mathbf{a}$

Example: $\langle -2, 3 \rangle + \langle 4, 1 \rangle = \langle 2, 4 \rangle$

5.3.1 The Parallelogram Law of Vector Addition

• Vectors $\mathbf{a}$ and $\mathbf{b}$ are represented as adjacent sides of a parallelogram; their sum $\mathbf{a} + \mathbf{b}$ is the diagonal from the common tail.

5.3.2 The Triangle Law of Vector Addition

• Place the tail of $\mathbf{b}$ at the head of $\mathbf{a}$; the vector from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$ is $\mathbf{a} + \mathbf{b}$.

5.4 Scalar Multiplication of Vectors

• Given a scalar $c$ and vector $\mathbf{a} = \langle a_1, a_2 \rangle$:

$c\mathbf{a} = \langle c a_1, c a_2 \rangle$

• The magnitude is:

$\|c\mathbf{a}\| = |c| \|\mathbf{a}\|$

• If $c > 0$, direction is unchanged; if $c < 0$, direction is reversed.

Example: $2\langle 3, 0 \rangle = \langle 6, 0 \rangle$; $(-1)\langle 3, 0 \rangle = \langle -3, 0 \rangle$

5.4.1 Unit Vectors

• A unit vector has magnitude 1.

• To normalize $\mathbf{a}$, divide by its magnitude:

$\hat{\mathbf{a}} = \frac{\mathbf{a}}{\|\mathbf{a}\|}$

• Example: For $\mathbf{a} = \langle 4, 2 \rangle$, $\|\mathbf{a}\| = \sqrt{20} = 2\sqrt{5}$, so $\hat{\mathbf{a}} = \langle \frac{4}{2\sqrt{5}}, \frac{2}{2\sqrt{5}} \rangle = \langle \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}} \rangle$.

5.5 Basis Vectors

• Any vector $\mathbf{a} = \langle a_1, a_2 \rangle$ can be written as:

$\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j}$

• Where $\mathbf{i} = \langle 1, 0 \rangle$ and $\mathbf{j} = \langle 0, 1 \rangle$ are the standard basis (unit) vectors.

• Components $a_1$ and $a_2$ are the scalar multiples of the basis vectors.

Example: If the tail of $\mathbf{a}$ is at A($-1$, 1) and the head at B(3, 3), then $a_1 = 4$, $a_2 = 2$, so $\mathbf{a} = 4\mathbf{i} + 2\mathbf{j}$.

5.6 The Difference of Two Vectors

• The difference is defined as:

$\mathbf{a} - \mathbf{b} = \langle a_1 - b_1, a_2 - b_2 \rangle$

• Equivalent to adding $\mathbf{a}$ and $-\mathbf{b}$.

Example: $\langle -2, 3 \rangle - \langle 4, 1 \rangle = \langle -6, 2 \rangle$

Summary Table: Vector Operations

Operation	Algebraic Rule	Geometric Interpretation
Addition	$\langle a_1, a_2 \rangle + \langle b_1, b_2 \rangle = \langle a_1 + b_1, a_2 + b_2 \rangle$	Parallelogram or triangle law
Scalar Multiplication	$c\langle a_1, a_2 \rangle = \langle c a_1, c a_2 \rangle$	Stretches or reverses vector
Magnitude	$\|\langle a_1, a_2 \rangle\| = \sqrt{a_1^2 + a_2^2}$	Length of arrow
Unit Vector	$\hat{\mathbf{a}} = \frac{\mathbf{a}}{\|\mathbf{a}\|}$	Direction only, length 1
Difference	$\langle a_1, a_2 \rangle - \langle b_1, b_2 \rangle = \langle a_1 - b_1, a_2 - b_2 \rangle$	Vector from tip of $\mathbf{b}$ to tip of $\mathbf{a}$

Additional info: These foundational concepts in vectors and the Cartesian plane are essential prerequisites for calculus topics such as limits, derivatives, and integrals, as they provide the geometric and algebraic framework for describing functions and their rates of change.